1 Revision 1 2 Optical absorption anisotropy of high-density, wide-gap, high-hardness SiO2 3 polymorphs seifertite, stishovite and coesite. 4 5 K. Kliera, J.A. Spirkob and K. M. Landskrona 6 7 a Department of Chemistry, Lehigh University, E. Packer Ave, Bethlehem, PA 18015 8 b Department of Physical and Environmental Sciences, Texas A&M University-Corpus 9 Christi, 6300 Ocean Dr, Unit 5802, Corpus Christi, TX 78414-5802 10 11 KEYWORDS: Theory, silica polymorphs, seifertite, stishovite, coesite, rutile, band 12 structure, effective mass, optical absorption 13 14 15 ABSTRACT 16 17 Dense, high-refractive index, ultra-hard, wide-gap polymorphs of SiO2, recently 18 discovered orthorhombic seifertite (space group 60 Pbcn), and earlier characterized 19 tetragonal stishovite (space group 136 P42/mnm) and monoclinic coesite (space group 15 1 20 C2/c) were studied using advanced methods of electronic structure calculations involving 21 full-potential linearized augmented plane wave density functional theoretical method 22 (FP-LAPW-DFT) with spin polarization, orbital dependent potentials, and modified 23 Becke-Johnson potential (mBJ) for accurate account of the band gaps. Even though these 24 calculations yield an excellent account of many properties, we here focus on quantitative 25 aspects of optical absorption and selection rules therein. Specifically, the valence-to- 26 conduction band transition in seifertite is symmetry-allowed, and is symmetry-forbidden 27 in stishovite and coesite. Theory is compared with published experimental data, and 28 explanation is provided for weak pre-edge optical absorption in stishovite. Electronic 29 structure and calculated properties of stishovite are also compared with those of the 30 isostructural rutile TiO2. Effective masses are calculated from the energy dispersion 31 curves E(k) at the valence band maximum for holes and conduction band minimum for 32 electrons. In addition, we propose that splitting of the O2p valence-band in coesite and 33 also observed in α-quartz is a general feature of polymorphs with tetrahedrally 34 coordinated Si, in contrast with continuous valence bands in those with octahedral SiO6 35 units such as seifertite and stishovite. Based on quantitative results obtained from the 36 BVA theory, this difference originates from a high degree of covalence in the tetrahedral 37 polymorphs as opposed to high iconicity in octahedral polymorphs. 38 39 I. INTRODUCTION 40 2 41 Dense, high-refractive index, ultra-hard, wide-gap polymorphs of SiO2 are of a great 42 interest not only owing to the history of discovery in extraterrestrial matter and their 43 extraordinary physical properties, but also to the underlying chemical bonding and 44 unusual coordination of the Si-On structure-forming units and their arrangements. The 45 most recently discovered and characterized polymorph of this type is orthorhombic 46 seifertite (space group 60 Pbcn) (Dera et al. 2002; El Goresy et al. 2008), after tetragonal 47 stishovite (space group 136 P42/mnm) was first synthesized in laboratory by Stishov and 48 Popova (1961) and reported on its natural occurrence in Arizona meteor crater by Chao et 49 al. (1962). The monoclinic coesite (space group 15 C2/c) was first synthesized by Coes 50 (1953), its natural occurrence reported by Chao et al. (1960) and crystal structure 51 determined by Levien and Prewitt (1981) and Smyth et al. (1987). Some of the 52 experimentally determined properties of the investigated silicas as well as of selected 53 reference materials are summarized in Table 1. 54 55 Table 1 56 57 The importance of these materials goes beyond their pure forms, especially as they serve 58 as matrices holding impurities which impart on them new properties, both in nature and 59 technology. For example, the exchange of 4H+ for Si4+, “the hydrogarnet substitution”, is 60 considered to be one of the mechanisms for hydrogen storage in the deep Earth, stishovite 61 being one of the vehicles for this storage as reported by Pawley et al. (1993), Williams 62 and Hemley (2001), the crystal structure of synthetic H-bearing aluminous stishovite 3 63 having been reported by Smyth et al. (1995). Related to technology, diffusion of Cu+ in 64 α-cristobalite was studied theoretically for possible implications to the functioning of 65 nanoelectronic devices by Zelený et al. (2012), prompting an interest in general 66 interaction of metals with the silica polymorphs. To advance the understanding of metal- 67 support interactions at surfaces related to adhesion and catalysis, strength of bonding and 68 agglomeration of Co and Ni on silica surfaces have been examined theoretically by Ma et 69 al. (2000), (2001a), compared with those on alumina support by Ma et al. (2001b), and 70 analyzed for periodic trends in interactions of the entire first-row transition metal series 71 with the silica surface by Ma et al. (2002) on a slab model derived from β-cristobalite, 72 using methods similar to those employed here, albeit focusing only on the ground-state 73 properties. 74 75 Theory has now advanced to a level of high reliability and accuracy to attack both the 76 fundamentals of electronic structure of pure crystalline materials such as the silicas 77 considered here, and the effects of impurities and deliberately added dopants on 78 properties such as optical spectra, luminescence, and carrier transport phenomena. While 79 electronic structure of stishovite and coesite has been addressed in numerous papers (Xu 80 and Ching 1991; Rudra and Fowler 1983), the most recently characterized seifertite has 81 not so far to our knowledge been subject to theoretical analysis. Experimental optical 82 absorption spectra of stishovite and coesite, but not seifertite, have also been reported by 83 Trukhin et al. (2004). In this work, we present calculations of all three high-pressure 84 polymorphs utilizing the full-potential linearized augmented plane wave (FP-LAPW) 85 method as described by Singh (1994a) and implemented by Blaha et al. (2013) with the 4 86 modified Becke-Johnson (mBJ) potential of Tran and Blaha (2009) that is suited for 87 analysis of the entire electronic structure including core levels, valence and conduction 88 bands involving both oxygen and silicon orbitals, accurate band gaps, and core-level 89 shifts (CLS) for interpretation of photoelectron spectra. Because stishovite is 90 isostructural with TiO2 rutile, we also compare their calculated properties and point out 91 the differences in bandgaps, optical transitions and carrier effective masses due to Ti ↔ 92 Si replacement. Reliability of the present FP-LAPW-mBJ method is tested by a 93 comparison with experiment of Garvie et al (2000) and with published calculations 94 employing the GW quasiparticle approximation for α-quartz by Chang et al. (2000). 95 Optical absorption spectra are calculated in the single-particle approximation of 96 Ambrosch-Draxl and Sofo (2006), and spin-orbit interaction, although very small in the 97 materials studied, is also assessed using the second variational method as implemented by 98 Novák (2001). The present theoretical results may be considered as a background, or a 99 first stage, for the interpretation of experimental absorption spectra, as previously done 100 for amorphous SiO2 using temperature dependence of Kramers-Kronig derived 101 absorption spectra in a recent study of Vella et al. (2011), as well as with theoretical 102 treatments of excitons in the pure and impurity-containing materials of this type, 103 employing methods already used for other SiO2 polymorphs: amorphous silica modeled 104 as β-cristobalite by Laughlin (1980), α-quartz SiO2 and TiO2 using the Bethe-Salpeter 105 equation (BSE) employed by Lawler et al. (2008) and Kang and Hybertsen (2010), and β- 106 cristobalite with quasi-particle corrections to the Kohn-Sham eigenvalues determined by 107 the GW approximation used by Ramos et al. (2004). 108 5 109 II. SEMI-EMPIRICAL ASSESSMENT OF BOND STRENGTHS, COVALENCE 110 AND IONICITY 111 112 An initial assessment of bonding and physical properties is motivated by the expediency 113 of semi-empirical methods for a large number of compounds, herein with emphasis on 114 the relation between known structures and degree of iconicity and covalence of the silica 115 polymorphs studied. While ionic compounds are stabilized by long-range electrostatic 116 [Madelung] interactions, covalent solids owe their stability to local bond strengths. The 117 silica polymorphs are expected to be an intermediate case, i.e. compounds that are 118 partially ionic and partially covalent. Presently we focus on the relation between 119 structures with octahedral and tetrahedral coordination of nearest-neighbor oxygen atoms 120 to Si and partial ionicity of the Si-O and Ti-O bonds. The simple semi-empirical analysis 121 presented here leads to a clear distinction between the more ionic, octahedrally 122 coordinated, and more covalent, tetrahedrally coordinated polymorphs. 123 124 One of the widely used semi-empirical analysis, cast in quantitative terms as the Bond- 125 Valence (BVA) theory summarized and reviewed in the recent book by Brown (2002), is 126 employed here in view of its great success in assaying the structure - bond strength 127 relationships for a large number of inorganic compounds as documented e.g. by Brese 128 and O’Keeffe (1991). Focusing on the MO2 oxides (M = Si, Ti), hexa- and tetra- 129 coordination of M atoms is linked to the ionicity or covalence of the M-O bond through 130 the bond strength as defined in the BVA method. The key relationship between bond 6 131 strength S and the M-O bond length R is formulated in terms of two empirical parameters, 132 the length R0 for a “reference” unit bond strength Sref = 1 and a gauge parameter b, as 133 = − 134 Sexp[( R0 R ) b ] (II-1). 135 136 In the present work, we used values R0 = 1.624 Ǻ for Si and 1.815 Ǻ for Ti, and b = 0.37 137 Ǻ for both Si and Ti in Si-O and Ti-O bonds.